David M. Bressoud

Macalester College, St. Paul, Minnesota

MathFest, Albuquerque, NM

August 5, 2005

MATH 136 DISCRETE MATHEMATICS

An introduction to the basic techniques and methods used in combinatorial problem-solving. Includes basic counting principles, induction, logic, recurrence relations, and graph theory. Every semester.

Required for a major or minor in Mathematics and in Computer Science.

I teach it as a project-driven course in combinatorics & number theory. Taught to 74 students, 3 sections, in 2004–05. More than 1 in 6 Macalester students take this course.

“Let us teach guessing” MAA video, George Pólya, 1965

Points:

•Difference between wild and educated guesses

•Importance of testing guesses

•Role of simpler problems

•Illustration of how instructive it can be to discover that you have made an incorrect guess

“Let us teach guessing” MAA video, George Pólya, 1965

Points:

•Difference between wild and educated guesses

•Importance of testing guesses

•Role of simpler problems

•Illustration of how instructive it can be to discover that you have made an incorrect guess Preparation:

•Some familiarity with proof by induction

•Review of binomial coefficients

Problem: How many regions are formed by 5 planes in space?

Start with wild guesses: 10, 25, 32, …

Problem: How many regions are formed by 5 planes in space?

Start with wild guesses: 10, 25, 32, …

random

Simpler problem:

0 planes: 1 region

1 plane: 2 regions

2 planes: 4 regions

3 planes: 8 regions

4 planes: ???

Problem: How many regions are formed by 5 planes in space?

Start with wild guesses: 10, 25, 32, …

random

Problem: How many regions are formed by 5 planes in space?

Simpler problem:

0 planes: 1 region

1 plane: 2 regions

2 planes: 4 regions

3 planes: 8 regions

4 planes: ???

Start with wild guesses: 10, 25, 32, …

Educated guess for 4 planes: 16 regions

random

TEST YOUR GUESS

Work with simpler problem: regions formed by lines on a plane:

0 lines: 1 region

1 line: 2 regions

2 lines: 4 regions

3 lines: ???

TEST YOUR GUESS

Work with simpler problem: regions formed by lines on a plane:

0 lines: 1 region

1 line: 2 regions

2 lines: 4 regions

3 lines: ???

1

23

4

5

6

7

START WITH SIMPLEST CASE

USE INDUCTIVE REASONING TO BUILD

n Space cut by n planes

Plane cut by n lines

Line cut by n points

0 1 1 1

1 2 2 2

2 4 4 3

3 8 7 4

4 5

5 6

START WITH SIMPLEST CASE

USE INDUCTIVE REASONING TO BUILD

n Space cut by n planes

Plane cut by n lines

Line cut by n points

0 1 1 1

1 2 2 2

2 4 4 3

3 8 7 4

4 11 5

5 6Test your guess

START WITH SIMPLEST CASE

USE INDUCTIVE REASONING TO BUILD

n Space cut by n planes

Plane cut by n lines

Line cut by n points

0 1 1 1

1 2 2 2

2 4 4 3

3 8 7 4

4 15 11 5

5 6Test your guess

GUESS A FORMULA

n points on a line

lines on a plane

planes in space

0 1 1 1

1 2 2 2

2 3 4 4

3 4 7 8

4 5 11 15

5 6 16 26

6 7 22 42

GUESS A FORMULA

n points on a line

lines on a plane

planes in space

0 1 1 1

1 2 2 2

2 3 4 4

3 4 7 8

4 5 11 15

5 6 16 26

6 7 22 42

0 1 2 3 4 5 6

0 1 0 0 0 0 0 0

1 1 1 0 0 0 0 0

2 1 2 1 0 0 0 0

3 1 3 3 1 0 0 0

4 1 4 6 4 1 0 0

5 1 5 10 10 5 1 0

6 1 6 15 20 15 6 1

k

n

n

k

⎛⎝⎜

⎞⎠⎟

GUESS A FORMULA

n k–1-dimensional hyperplanes in k-dimensional space cut it into:

n

0

⎛⎝⎜

⎞⎠⎟+

n1

⎛⎝⎜

⎞⎠⎟+

n2

⎛⎝⎜

⎞⎠⎟+L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

GUESS A FORMULA

n

0

⎛⎝⎜

⎞⎠⎟+

n1

⎛⎝⎜

⎞⎠⎟+

n2

⎛⎝⎜

⎞⎠⎟+L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

Now prove it!

n k–1-dimensional hyperplanes in k-dimensional space cut it into:

GUESS A FORMULA

n

0

⎛⎝⎜

⎞⎠⎟+

n1

⎛⎝⎜

⎞⎠⎟+

n2

⎛⎝⎜

⎞⎠⎟+L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

Now prove it!

Show that if R n,k( ) =# of regions with n hyperplanes

in k-dimensional space, then

R(n,k) =R(n−1,k) + R(n−1,k−1).What do you have to assume about k−1-hyperplanes in k-dimensional space?

n k–1-dimensional hyperplanes in k-dimensional space cut it into:

This PowerPoint presentation and the Project Description are available at

www.macalester.edu/~bressoud/talks